4 NOTE ON BESSEL'S FUNCTIONS AS APPLIED since by (3) » n) + n n = - n In (5) all the denominators are positive. We deduce „ 2 _ ™2 ,?• 2 _ OT2 -. 2 _ ™2 gl ^ __ 1 , *1 _ 11 I ^ ^ , ^ I • * and therefore z? >n2 + '2n>n(n + 1). Our theorems are therefore proved. If a closer approximation to z-f is desired, it may be obtaine stituting on the right of (6) 2n for z? — n2 in the numerators and i n2 in the denominators. Thus r\ -•' J- T •*•"" 2n •^ + 2« ,,«! , «2 « *s i ... ,„/»,~L~O\ r • ft ^l +• 4; J Now, as is easily proved from the ascending series for Jn', 71+2 ty ~"2 _J_ /y "™*2 1_ «• "~~2 _l — • so that finally w3 z? >n-+ 2-» +,-— When n is very great, it will follow from (7) that z? > v/,2 + the approximation is not close, for the ultimate form is* ^ = 7is+[1-6130] n4* As has been mentioned, the sequence formula In Jn (g) = Jn_1 (z] + J n+l prohibits the simultaneous evanescence of J"n_1 and Jn, or of Jn-i The question arises—can Bessel's functions whose orders (supposei differ by more than 2 vanish simultaneously ? If we change n in (8) and then eliminate Jn, we get _ - "«— 1 ~1 '" tl+2> from which it appears that if Jn^ and Jn+2 vanish simultaneously, t Jn+i = 0, which is impossible, or ^-' = 4<n (n + 1). Any common rt and Jn+i must therefore be such that its square is an integer. * Phil. Mag. Vol. xx. p. 1003, 1910, equation (8). [1913. A correction is hen See Nicholson, Phil. Mag. Vol. xxv. p. 200, 1913.]emains to prove that z* necessarily exceeds n(n 4-1). That zli exceeds n2 is well known*, but this does not suffice. We can obtain what we require from a formula given, in Theory of Sound, 2nd ed. § 339. If the finite roots taken in order be zi} zz,... za..., we may write